There are many reasons why there was a quote that says "Glory that was Greece". One reason could be because of the great influences and contributions Greeks have made to the modern civilization. Here are some of the listed reasons below. I hope it would be helpful to you. God Bless You!
The Greek Mythology

"The Great Altar Of Zeus"
Although the Greek mythology could be considered to be as an idolatry, we could never hide the fact that the beautiful temples and basilicas made for the gods are unsurpassed. The beautiful stories of wars and love wrote by the poet Homer is incomparable,why? because although he was blind, he was able to narrate the story with such passion and persuasion. During the invasion of the Romans in Greece, instead of influencing the Greeks, they adapted the culture. The Romans adapted the Greek gods and made their equivalents.
The Greatest Art

image source: Wikipedia
For the Greeks, the man's body is the greatest piece of artwork because it is made by none other than the gods themselves. Humans can do tremendous things which other species cannot. There are also some Greek arts such as the imprinted symbols on pots .
The Greatest Poetry, Comedy, Playwright

"The statue of Euripides"
Among any other writers, the Greek always stand out because of their writing prowess. Just talking about the works of Aeschylus, Euripides and Sophocles, you cannot even say that you have attained half of their achievements already. Even Edgar Allan Poe cannot be compared to the Greek writers because they are not in the same level.
The Origin Of Modern Day Terms

Most of the simple and medical terms we know now did really came from the Greeks. That is why most of the words that are in the dictionary have there origin being traced up to the ancient Greek language.
The Greatest Philisophers

"The Famous Philosopher: Plato"
The greatest Philosophers are to be found in Greece. Their knowledge in everything is very great. Although they also adapted some of the cultures of different civilizations, you could not say that they plainly copied. Although they adapted, they were able modify and make the tradition even better that made it as their own.
The Traditional Medicine

Do you know that even though America has the greater advancement of technology, it is in Greece wherein the observance of patients began. The very first person to formulate the anatomical work was Alcmaeon. The Greek way of treating patients even reached the time of the Romans because of the practice were absorbed by them. The doctrine of medicinal practice that was being followed was the work of Hippocrates which was the Hippocatric Medicine.